probability – Clarification regarding server times and conditional expectation.

Suppose there are servers $S_1,S_2,S_3$ such that they have service times following an exponential distribution with rates $mu_1,mu_2,mu_3$, respectively.

Upon entering the system a person must be serviced in the following order : $S_1 to S_2 to S_3$

Consider the scenario where once we arrive there is only one customer ahead of us at $S_3$.

I am in need of a clarification for the expected time in the system given $S_3$ is occupied when we arrive to use it (we are under the assumption if a server is busy we must wait for it to clear the current user).

Using a prepared solution we get the following:
Let $T:= $ time in system.

$displaystyle E(T mid S_3 text{ is occupied}) = bigg(frac{mu_1}{mu_1+mu_3}bigg)_{(1)}cdotbigg(frac{mu_2}{mu_1+mu_3}bigg)_{(2)}bigg(frac{1}{mu_3}bigg)_{(3)}$

I have used subscripts to label each term.
Now, $(1)$ makes sense as this is service time at $S_1$ being less than the service time at $S_3$. Additionally, $(3)$ makes sense as this is expected service time at $S_3$.

The clarification I am after is for $(2)$, it seems we would want $displaystyle frac{mu_2}{mu_2+mu_3}$ because this would be that the service time at $S_2$ is less than service time at $S_3$. However in the solution the denominator in $(2)$ is $mu_1+mu_2$.

Is the solution correct? If so can you provide an explanation as why $(2)$ has this form?

conditional formatting – How do I highlight a cell based on time?

For example, I want to have a cell highlighted while an employee is on break. So for this I have an “OUT” time and an “IN” time. If the employee goes on break at 14:22 and is back at 14:37, I want 14:37 to be highlighted red until that person is due back

Is there a way (and if so how) to add a conditional trait into PCGen?

Specifically what I am trying to do is add a racial trait to the Kitsune race that will replace two of the normal racial traits. I have successfully added a trait called “Naturally Skilled” into the program and it shows up/ works just fine however I need to cut out two of the other traits you would get naturally if you choose this trait.

Starting racial traits for a Kitsune:
Agile, Change Shape, Kitsune Magic, Low-Light Vision

What it should look like if I choose Naturally Skilled is:
Change Shape, Low-Light Vision, Naturally Skilled

So essentially I’m replacing both Agile and Kitsune Magic for Naturally Skilled. Is there a good way to code that in? I’ve been trying to find where in the code to put arguments like that but its like sifting through a barn full of hay for a hay colored needle.

Google Sheets. How to combine a checkbox and conditional formatting?

Here’s my problem.

I have a cell with a checkbox. I can make it turn green when I check it. But I want it to turn green if no conditions are met and turn orange when a condition is met. The condition is that a range of four cells is empty (<>""). If all of them are empty – then checkbox becomes orange when pressed, if there is anything in any of the four cell then checkbox becomes green when pressed.

microsoft excel – Apply conditional formatting on different column as the column with the conditional formula

I have an excel table that looks like this (but bigger).

enter image description here

If the values in the 2nd, 4th and 6th column are bigger than 0.25, I want the value in the previous column to be red with conditional formatting, like so:

enter image description here

I got pretty far with this question
Conditional formatting based on several columns
using formula =$C$4>0,25 applied to =$B$4
But this only works for one cell. How do I apply this to the entire file?

probability theory – Conditional expectations of $X$ with respect to a filtration converges to $X$

Let $mathbb F = left( mathcal F_nright)_{n in mathbb N}$ be a filtration, and let $mathcal F_infty = sigmaleft(mathcal F_n : n geq 1right)$. I want to try proving the following result:

If $X in mathcal L^1left(mathcal F_inftyright)$, then $mathbb Eleft(X|mathcal F_nright) to X$ as $n to infty$ almost surely.

It’s easy to show that $left(mathbb Eleft(X|mathcal F_nright)right)_{n in mathbb N}$ is a martingale with respect to $mathbb F$, and since $X$ is integrable, one can show that
sup_{n geq 1} mathbb Eleft(mathbb Eleft(X|mathcal F_nright)^+right) leq mathbb E(|X|) < infty

So by the martingale convergence theorem, $left(mathbb Eleft(X|mathcal F_nright)right)_{n geq 1}$ converges almost surely, say to the random variable $tilde X$. But why does $X = tilde X$ a.s.? How can I show that $left| X – mathbb Eleft(X|mathcal F_nright)right| to 0$? I’m wondering if there’s a functional analytic reason, considering the projections $mathbb Eleft(cdot|mathcal F_nright) : mathcal L^1left(mathcal F_inftyright) to mathcal L^1left(mathcal F_nright)$, but I haven’t gotten very far with this approach. Or is there an obvious reason for this that I’m just overlooking?

column formatting – Conditional Popup

I’ve created a custom view that includes a popup description of each file in my sharepoint library. This works well when I’m within a sub-folder that has a list of files, but when I navigate to the main list of sub-folders within the library the custom formatting continues and a blank pop up appears because the folders don’t have descriptions. Is there a way to use conditional formatting so that the pop up only appears for rows that have text in the description column?

This is my code currently:

  "elmType": "div",
  "style": {
    "font-size": "16px",
    "color": "white"
  "txtContent": "@currentField",
  "customCardProps": {
    "formatter": {
      "elmType": "div",
      "txtContent": "[$Description]",
      "style": {
        "font-size": "14px",
        "padding": "10px",
        "width": "500px",
        "color": "black",
        "background-color": "white"
    "openOnEvent": "hover",
    "directionalHint": "bottomCenter",
    "isBeakVisible": true,
    "beakStyle": {
      "backgroundColor": "white"


google sheets – I want a custom conditional formatting to fill a color for a cell depending on the number value after the decimal point

I am entering both positive and negative numbers in a column and want a way to identify them without an extra column for example -100.03 or -250.1 or 100.015. in this example I am listing moneyline betting odds and want to use the .1 to identify inside distance and .03 to mean over 3 rounds, and .015 to mean over 1.5 rounds. This will give me the type of prop bet even though i just have the number entered without the need for an extra column to identify it. I was thinking about regular expressions but not sure

How to elevate a conditional format rule to be the number one rule instantly

I would like to make a newly created rule for conditional formatting the top rule so it has precedent over all others but be able to do so without having to drag it to the top as this takes for ever if the column has many rules like 30 to 40 already in place for different cells within the column.

Is independence preserved under conditional expectations?

Suppose $X,Y,Z$ are random variables and $Y$ is independent of $Z$. Is $E[Y|X]$ independent of $Z$? In other words, can the conditional expectation operation "erase" independence? Of course, $X$ and $Z$ need not be independent.